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## G = C2×C33⋊D4order 432 = 24·33

### Direct product of C2 and C33⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×C3⋊S3 — C2×C33⋊D4
 Chief series C1 — C3 — C33 — C3×C3⋊S3 — C32⋊4D6 — C33⋊D4 — C2×C33⋊D4
 Lower central C33 — C3×C3⋊S3 — C2×C33⋊D4
 Upper central C1 — C2

Generators and relations for C2×C33⋊D4
G = < a,b,c,d,e,f | a2=b3=c3=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=c-1, fbf=c, cd=dc, ece-1=fcf=b, ede-1=fdf=d-1, fef=e-1 >

Subgroups: 1396 in 192 conjugacy classes, 31 normal (19 characteristic)
C1, C2, C2 [×6], C3, C3 [×4], C4 [×2], C22 [×9], S3 [×12], C6, C6 [×14], C2×C4, D4 [×4], C23 [×2], C32, C32 [×4], Dic3 [×2], D6 [×17], C2×C6 [×8], C2×D4, C3×S3 [×16], C3⋊S3 [×2], C3⋊S3 [×2], C3×C6, C3×C6 [×6], C2×Dic3, C3⋊D4 [×4], C22×S3 [×3], C22×C6, C33, C32⋊C4 [×2], S32 [×2], S32 [×8], S3×C6 [×12], C2×C3⋊S3, C2×C3⋊S3, C62, C2×C3⋊D4, S3×C32 [×2], C3×C3⋊S3 [×2], C3×C3⋊S3 [×2], C32×C6, S3≀C2 [×4], C2×C32⋊C4, C2×S32, C2×S32 [×2], S3×C2×C6, C33⋊C4 [×2], C3×S32 [×2], C3×S32, C324D6 [×2], C324D6, S3×C3×C6, C6×C3⋊S3, C6×C3⋊S3, C2×S3≀C2, C33⋊D4 [×4], C2×C33⋊C4, S32×C6, C2×C324D6, C2×C33⋊D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C3⋊D4 [×2], C22×S3, C2×C3⋊D4, S3≀C2, C2×S3≀C2, C33⋊D4, C2×C33⋊D4

Permutation representations of C2×C33⋊D4
On 24 points - transitive group 24T1292
Generators in S24
(1 17)(2 18)(3 19)(4 20)(5 15)(6 16)(7 13)(8 14)(9 21)(10 22)(11 23)(12 24)
(2 6 10)(4 12 8)(14 20 24)(16 22 18)
(1 9 5)(3 7 11)(13 23 19)(15 17 21)
(1 9 5)(2 6 10)(3 11 7)(4 8 12)(13 19 23)(14 24 20)(15 17 21)(16 22 18)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 20)(2 19)(3 18)(4 17)(5 14)(6 13)(7 16)(8 15)(9 24)(10 23)(11 22)(12 21)

G:=sub<Sym(24)| (1,17)(2,18)(3,19)(4,20)(5,15)(6,16)(7,13)(8,14)(9,21)(10,22)(11,23)(12,24), (2,6,10)(4,12,8)(14,20,24)(16,22,18), (1,9,5)(3,7,11)(13,23,19)(15,17,21), (1,9,5)(2,6,10)(3,11,7)(4,8,12)(13,19,23)(14,24,20)(15,17,21)(16,22,18), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,20)(2,19)(3,18)(4,17)(5,14)(6,13)(7,16)(8,15)(9,24)(10,23)(11,22)(12,21)>;

G:=Group( (1,17)(2,18)(3,19)(4,20)(5,15)(6,16)(7,13)(8,14)(9,21)(10,22)(11,23)(12,24), (2,6,10)(4,12,8)(14,20,24)(16,22,18), (1,9,5)(3,7,11)(13,23,19)(15,17,21), (1,9,5)(2,6,10)(3,11,7)(4,8,12)(13,19,23)(14,24,20)(15,17,21)(16,22,18), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,20)(2,19)(3,18)(4,17)(5,14)(6,13)(7,16)(8,15)(9,24)(10,23)(11,22)(12,21) );

G=PermutationGroup([(1,17),(2,18),(3,19),(4,20),(5,15),(6,16),(7,13),(8,14),(9,21),(10,22),(11,23),(12,24)], [(2,6,10),(4,12,8),(14,20,24),(16,22,18)], [(1,9,5),(3,7,11),(13,23,19),(15,17,21)], [(1,9,5),(2,6,10),(3,11,7),(4,8,12),(13,19,23),(14,24,20),(15,17,21),(16,22,18)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,20),(2,19),(3,18),(4,17),(5,14),(6,13),(7,16),(8,15),(9,24),(10,23),(11,22),(12,21)])

G:=TransitiveGroup(24,1292);

On 24 points - transitive group 24T1312
Generators in S24
(1 6)(2 7)(3 8)(4 5)(9 19)(10 20)(11 17)(12 18)(13 24)(14 21)(15 22)(16 23)
(1 21 20)(2 22 17)(3 18 23)(4 19 24)(5 9 13)(6 14 10)(7 15 11)(8 12 16)
(1 20 21)(2 22 17)(3 23 18)(4 19 24)(5 9 13)(6 10 14)(7 15 11)(8 16 12)
(1 20 21)(2 22 17)(3 18 23)(4 24 19)(5 13 9)(6 10 14)(7 15 11)(8 12 16)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 6)(2 5)(3 8)(4 7)(9 22)(10 21)(11 24)(12 23)(13 17)(14 20)(15 19)(16 18)

G:=sub<Sym(24)| (1,6)(2,7)(3,8)(4,5)(9,19)(10,20)(11,17)(12,18)(13,24)(14,21)(15,22)(16,23), (1,21,20)(2,22,17)(3,18,23)(4,19,24)(5,9,13)(6,14,10)(7,15,11)(8,12,16), (1,20,21)(2,22,17)(3,23,18)(4,19,24)(5,9,13)(6,10,14)(7,15,11)(8,16,12), (1,20,21)(2,22,17)(3,18,23)(4,24,19)(5,13,9)(6,10,14)(7,15,11)(8,12,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,6)(2,5)(3,8)(4,7)(9,22)(10,21)(11,24)(12,23)(13,17)(14,20)(15,19)(16,18)>;

G:=Group( (1,6)(2,7)(3,8)(4,5)(9,19)(10,20)(11,17)(12,18)(13,24)(14,21)(15,22)(16,23), (1,21,20)(2,22,17)(3,18,23)(4,19,24)(5,9,13)(6,14,10)(7,15,11)(8,12,16), (1,20,21)(2,22,17)(3,23,18)(4,19,24)(5,9,13)(6,10,14)(7,15,11)(8,16,12), (1,20,21)(2,22,17)(3,18,23)(4,24,19)(5,13,9)(6,10,14)(7,15,11)(8,12,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,6)(2,5)(3,8)(4,7)(9,22)(10,21)(11,24)(12,23)(13,17)(14,20)(15,19)(16,18) );

G=PermutationGroup([(1,6),(2,7),(3,8),(4,5),(9,19),(10,20),(11,17),(12,18),(13,24),(14,21),(15,22),(16,23)], [(1,21,20),(2,22,17),(3,18,23),(4,19,24),(5,9,13),(6,14,10),(7,15,11),(8,12,16)], [(1,20,21),(2,22,17),(3,23,18),(4,19,24),(5,9,13),(6,10,14),(7,15,11),(8,16,12)], [(1,20,21),(2,22,17),(3,18,23),(4,24,19),(5,13,9),(6,10,14),(7,15,11),(8,12,16)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,6),(2,5),(3,8),(4,7),(9,22),(10,21),(11,24),(12,23),(13,17),(14,20),(15,19),(16,18)])

G:=TransitiveGroup(24,1312);

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 3F 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K ··· 6P 6Q 6R 6S 6T order 1 2 2 2 2 2 2 2 3 3 3 3 3 3 4 4 6 6 6 6 6 6 6 6 6 6 6 ··· 6 6 6 6 6 size 1 1 6 6 9 9 18 18 2 4 4 4 4 8 54 54 2 4 4 4 4 6 6 6 6 8 12 ··· 12 18 18 36 36

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 8 8 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D4 D6 D6 C3⋊D4 C3⋊D4 S3≀C2 C2×S3≀C2 C33⋊D4 C2×C33⋊D4 C33⋊D4 C2×C33⋊D4 kernel C2×C33⋊D4 C33⋊D4 C2×C33⋊C4 S32×C6 C2×C32⋊4D6 C2×S32 C3×C3⋊S3 C32×C6 S32 C2×C3⋊S3 C3⋊S3 C3×C6 C6 C3 C2 C1 C2 C1 # reps 1 4 1 1 1 1 1 1 2 1 2 2 4 4 4 4 1 1

Matrix representation of C2×C33⋊D4 in GL4(𝔽7) generated by

 6 0 0 0 0 6 0 0 0 0 6 0 0 0 0 6
,
 1 0 4 0 5 6 1 4 4 4 0 6 0 0 0 1
,
 5 3 5 3 3 5 2 3 0 0 1 0 0 0 0 4
,
 3 1 4 5 1 3 3 5 0 0 4 0 0 0 0 2
,
 6 2 6 5 1 2 1 4 5 2 1 5 1 1 3 5
,
 5 3 3 4 5 5 6 1 2 5 6 2 1 1 3 5
G:=sub<GL(4,GF(7))| [6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[1,5,4,0,0,6,4,0,4,1,0,0,0,4,6,1],[5,3,0,0,3,5,0,0,5,2,1,0,3,3,0,4],[3,1,0,0,1,3,0,0,4,3,4,0,5,5,0,2],[6,1,5,1,2,2,2,1,6,1,1,3,5,4,5,5],[5,5,2,1,3,5,5,1,3,6,6,3,4,1,2,5] >;

C2×C33⋊D4 in GAP, Magma, Sage, TeX

C_2\times C_3^3\rtimes D_4
% in TeX

G:=Group("C2xC3^3:D4");
// GroupNames label

G:=SmallGroup(432,755);
// by ID

G=gap.SmallGroup(432,755);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,141,1684,571,165,677,530,14118]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,e*b*e^-1=c^-1,f*b*f=c,c*d=d*c,e*c*e^-1=f*c*f=b,e*d*e^-1=f*d*f=d^-1,f*e*f=e^-1>;
// generators/relations

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